The paper introduces Atiyah classes for SH Lie pairs and modules.
problem Measuring nontriviality of SH Lie algebras and their extensions.
method Introduces Atiyah classes and studies their properties and invariance.
result Atiyah classes induce graded Lie algebra structures and Lie algebra module structures.
In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector …
Reduced sh-Lie structures have been studied for the case when a Lie group acts on the fibers of a vector bundle while preserving the base space of the bundle. In this paper we investigate how one obtains a reduced sh-Lie structure using the ideas of symmetry reduction where the action of the Lie group is transversal to…
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are con…
We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant Lie algebra. To this end we endow the homology H(g) of any differential grad…
After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
Constructs a bilinear form from a quasimorphism on symplectic manifold groups.
problem Understanding symplectic group properties through quasimorphisms and bilinear forms.
method Develops machinery to construct a real-valued bilinear form from a quasimorphism on the commutator subgroup of symplectic group.
result The constructed bilinear form b controls extendability of quasimorphisms and triviality of characteristic classes. In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac struct…
In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part …
Combination theorem for geodesic coarsely convex group pairs.
problem Understanding properties of groups relative to subgroups.
method Definitions of weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex group pairs; combination theorem.
result Combination theorem for geodesic coarsely convex group pairs.
Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …
The article introduces a new shape signature method using homology nerves and proximal relators.
problem Describing the shape of finite, bounded planar objects.
method Planar shape signatures derived from homology nerves with proximal relators.
result Every finite, bounded planar shape has a signature derived from the homology group.
Let $ψ:\M \to \SH$ be an isometric immersion of codimension 1, then there exist symmetric (1,1)-tensors S and f, a tangent vector field U and a smooth function λ on $\M$ that satisfy the compatibility equations of $\SH$. In this paper, we will deal with the converse problem: "Given a Riemannian manifold $\M$ …
Meta-learning improves hyperparameter tuning for XGBoost.
problem Improving hyperparameter tuning for XGBoost models.
method Proposed MeSH algorithm using meta-regressors to guide hyperparameter selection.
result MeSH often finds superior hyperparameter configurations compared to SH and random search.
Improved batched SH algorithm maintains original performance.
problem Maintaining performance in batched multi-armed bandits.
method Simple batch version of Sequential Halving algorithm.
result Batching does not degrade SH algorithm's performance.
Let (X,ω) be a compact Kähler manifold of dimension n and fix m∈N such that 1≤m≤n. We prove that any (ω,m)-sh function can be approximated from above by smooth (ω,m)-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluri…
The paper constructs structures for Lie pairs and their Atiyah classes.
problem Understanding structures of Lie pairs and their Atiyah classes.
method Constructs dg-manifolds and dg-Lie algebroids for Lie pairs, showing quasi-isomorphisms and Atiyah classes.
result Induces a quasi-isomorphism between dg-Lie algebroids and Atiyah classes of Lie pairs.
Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.
problem Understanding sheaf equivalences and actions on Lagrangian cobordisms.
method Analyzing sheaf quantizations and Legendrian lifts, proving functorial properties.
result Lagrangian cobordism functor is action decreasing and Morita equivalent to sheaf categories of Legendrians.
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
Hamiltonian dynamics on matched pairs groups are explored.
problem Exploring Hamiltonian dynamics on matched pairs Lie groups.
method Trivialization of cotangent bundles, explicit symplectic and Poisson brackets, reductions, Lie-Poisson bracket derivation.
result Lie-Poisson equations on sl(2,C)∗ are derived. Let X be a compact real analytic manifold, and let T∗X be its cotangent bundle. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X. In this paper, we develop a Fukaya A∞-category Fuk(T∗X) whose objects are exact, not necessarily compact Lagrangian branes in…
Geometrically interprets Atiyah class vanishing for Lie pairs.
problem Atiyah class vanishing for Lie pairs and Lie algebroids.
method New characterisation of ideal systems and geometric interpretation.
result Atiyah class vanishes for Lie pairs with kernel of fibration.
Two new equivalences found between Lie supergroups and super Harish-Chandra pairs.
problem Establishing a new equivalence between Lie supergroups and super Harish-Chandra pairs.
method Two new functors Ψ^circ and Ψ^e that construct a Lie supergroup from a super Harish-Chandra pair.
result Both Ψ^circ and Ψ^e are quasi-inverse to the natural functor Φ from Lie supergroups to super Harish-Chandra pairs.
Study of Lagrangian dynamics on matched Lie groups.
problem Understanding dynamics on matched Lie groups.
method Isomorphic tangent bundle and Euler-Lagrange/Poincaré equations.
result Covering semi-direct product theory and explicit equations.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of (L,A) modulo automorphisms from exponentials of derivations of L and those from the exponentials of inner derivations of L. result Find the associated governing L∞-algebras in the sense of extended deformation theory. We generalize the Schouten calculus of multivector fields to commutative Lie Rinehart pairs and define a non negatively graded Lie oo-algebra on their exterior power.
Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
New internal symmetry found for Lie pair algebra.
problem Understanding Lie pair structures and their associated algebras.
method Introduced a Lie algebra action by Der(L) on the L_{≤3} algebra.
result Found internal symmetry of the L_{≤3} algebra.
Analysis of Vlasov plasma dynamics using matched pair Lie-Poisson formulation.
problem Understanding the dynamics of Vlasov plasma and its kinetic moments.
method Hamiltonian (Lie-Poisson) analysis and matched pair decomposition.
result Observation of mutual interactions between subdynamics in Vlasov plasma.
Study uses Vinberg pairs for Higgs bundles, revealing their role.
problem Understanding the role of Vinberg pairs in Higgs bundle theory.
method Exploring Vinberg pairs defined by cyclic gradings of a Lie algebra in Higgs bundle theory.
result Vinberg pairs play a significant role in Higgs bundle theory.
The paper studies connections on Lie groupoid pairs and their linearization.
problem Linearizing actions on homogeneous spaces of Lie groupoid pairs.
method Develops a theory of connections on equivariant principal bundles, associates an Atiyah class, and uses it to show linearization conditions.
result Necessary and sufficient conditions for linearizing dressing actions and monodromies.
Integrates Lax pair equations for a specific Lie algebra.
problem Low-dimensional Lie algebras of infinitesimal character ildeβ0. method Shows complete integrability of Lax pair equations.
result Proves complete integrability for certain Lie algebras.
Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.
problem Vertical isomorphisms of Fedosov dg manifolds associated with Lie pairs.
method Construction of Fedosov dg manifolds via splitting and connection, proving unique isomorphisms using iteration formula.
result Existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds.
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
problem Constructing Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
method Using homological perturbation lemma and contraction, constructing isomorphisms between cochain complexes and Chevalley-Eilenberg cohomologies.
result Identifies Atiyah and Todd classes of Lie pair pullback dg Lie algebroids with those of the Lie pair.
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,∇) where B ⊂ A are regular Lie algebroids, both over the same regular foliated manifold (M,…
Examines Lie algebras for symmetries in odd-dimensional manifolds.
problem Symmetry analysis in odd-dimensional manifolds.
method Analyzes local Lie algebras of pairs of functions.
result Identifies infinitesimal symmetries of almost-cosymplectic-contact structures.
The paper presents new representations and spherical indicatrices of Bertrand curves in Lie groups.
problem Understanding geometric properties of Bertrand curves in Lie groups.
method New representations and spherical indicatrices of Bertrand curves in three Lie groups with bi-invariant metrics are derived.
result Relations between spherical indicatrices and new representations of Bertrand curves are established.
In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair (L,A) of algebroids. In particular, we prove that the quotient L/A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.
Geometrically explains Lie 2-algebroids and their connections.
problem Exploring Lie 2-algebroids and their geometric properties.
method Explains Li-Bland's correspondence and uses geometric equivalence.
result Proves bicrossproduct of matched pairs of 2-representations is a split Lie 2-algebroid.
We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the tangent of a Lie algebroid.
The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.
problem Exploring connections between Lie bialgebras and Rota-Baxter Lie algebras.
method Introducing quadratic Rota-Baxter Lie algebras, matched pairs, bialgebras, and Manin triples.
result Established a correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras.
New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
Paper classifies local moves and shows how they affect knot transformations.
problem Classifying local moves and understanding their impact on knot transformations.
method Introduced an equivalence relation and binary relation on local moves, and used these to analyze knot transformations.
result Discovered that certain extended ST-moves can unknot a knot.